This research protocol is a controlled pilot study of immunotherapeutic infusion of HIV-specific cytotoxic T lymphocytes (CTL) following combination antiviral drug therapy for asymptomatic HIV-infected subjects. The trial builds on small pilot trials conducted in HIV-infected subjects with more advanced disease in which therapeutic infusion of antiviral CTL was shown to be both safe and feasible. Our hypothesis is that excessive viral antigen may interfere with the proliferation, function, and continued survival of antiviral CD4 and CD8 T cells, which are the cornerstones of protective immunity. We will test whether CTL infusion will be effective at controlling viral replication when the viral burden is low and when the other arms of the immune system are still functional. The goal of the study will be to reduce circulating viral RNA levels to below detectable with antiviral drugs and to determine if the added infusion of anti-HIV CTL after achieving the expected viral nadir results in more rapid and complete immune restoration and more sustained reduction of viral burden in blood and lymph nodes.